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Space- and Time-Dependent Quantum Dynamics of Spatially Indirect Excitons in Semiconductor Heterostructures Federico Grasselli, Andrea Bertoni, and Guido Goldoni

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Space- and Time-Dependent Quantum Dynamics of Spatially Indirect Excitons in Semiconductor Heterostructures Federico Grasselli, Andrea Bertoni, and Guido Goldoni Space- and time-dependent quantum dynamics of spatially indirect excitons in semiconductor heterostructures Federico Grasselli, Andrea Bertoni, and Guido Goldoni Citation: The Journal of Chemical Physics 142, 034701 (2015); doi: 10.1063/1.4905483 View online: http://dx.doi.org/10.1063/1.4905483 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dynamics of indirect excitons in a coupled quantum-well pair J. Appl. Phys. 113, 153706 (2013); 10.1063/1.4801808 On Bose condensation of excitons in quasi-two-dimensional semiconductor heterostructures Low Temp. Phys. 38, 541 (2012); 10.1063/1.4733681 Tunneling escape time from a semiconductor quantum well in an electric field J. Appl. 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Downloaded to IP: 155.185.14.147 On: Thu, 15 Jan 2015 13:57:37 THE JOURNAL OF CHEMICAL PHYSICS 142, 034701 (2015) Space- and time-dependent quantum dynamics of spatially indirect excitons in semiconductor heterostructures Federico Grasselli,1,2,a) Andrea Bertoni,2,b) and Guido Goldoni1,2,c) 1Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia, Modena, Italy 2CNR-NANO S3, Institute for Nanoscience, Via Campi 213/a, 41125 Modena, Italy (Received 10 November 2014; accepted 22 December 2014; published online 15 January 2015) We study the unitary propagation of a two-particle one-dimensional Schrödinger equation by means of the Split-Step Fourier method, to study the coherent evolution of a spatially indirect exciton (IX) in semiconductor heterostructures. The mutual Coulomb interaction of the electron-hole pair and the electrostatic potentials generated by external gates and acting on the two particles separately are taken into account exactly in the two-particle dynamics. As relevant examples, step/downhill and barrier/well potential profiles are considered. The space- and time-dependent evolutions during the scattering event as well as the asymptotic time behavior are analyzed. For typical parameters of GaAs- based devices, the transmission or reflection of the pair turns out to be a complex two-particle process, due to comparable and competing Coulomb, electrostatic, and kinetic energy scales. Depending on the intensity and anisotropy of the scattering potentials, the quantum evolution may result in excitation of the IX internal degrees of freedom, dissociation of the pair, or transmission in small periodic IX wavepackets due to dwelling of one particle in the barrier region. We discuss the occurrence of each process in the full parameter space of the scattering potentials and the relevance of our results for current excitronic technologies. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4905483] I. INTRODUCTION several micrometers,7,12 while recombination can be controlled by switching off the perpendicular field,13 broadening their Excitonic states are the fundamental below-gap reso- application to an entirely new class of opto-electronic devices. nances in semiconductors heterostructures, where stability In fact, in these devices, data carried by photons can be stored and strength are strongly enhanced by quantum confinement.1 in IXs, elaborated as in charge-based nano-electronics, and Spatially indirect excitons (IXs) are photo-excited correlated finally released back to photons by switching off the perpendic- electron-hole pairs with the two charges localized in different ular electric field at the desired time, all processes being inte- segments of a nano-material. For example, IXs can be easily grated in the same semiconductor segment. Electronics based and selectively excited in semiconductor coupled quantum on excitons is called excitronics. Since gating can be modulated wells (CQWs), where an electric field applied perpendicular to at the GHz frequency, that is, orders of magnitude shorter than the plane of the quantum wells separates positive and negative the IXs intrinsic lifetime, excitronic systems set themselves charges in different layers.2–4 Since optical recombination as a natural bridge between optical data communication and depends on the electron-hole overlap, this extends the intrinsic nano-electronic devices.14–16 lifetime of excitons by orders of magnitude, from nanosec- Several excitronic functions have been demonstrated such onds5 to microseconds,6,7 which allows for the formation of as fast data storage,13,17 acceleration with electrostatic ramps7,18 new types of quantum condensates.6,8–10 and interdigital devices,19 and field effect transistors.12 Most In CQWs with an average inter-well separation d, IXs devices work in a diffusive regime, similar to traditional elec- are two-dimensional quasi-particles in, say, the x;y plane tronics. Therefore, IX is usually described theoretically as a carrying a permanent electric dipole of order −ed( along) the classical gas of interacting dipoles.19,20 However, trapping of growth direction z. Beyond any change in the binding energy single IXs has been recently demonstrated,21 opening to the implied by the particle separation, an electric field Fz shifts realization of quantum excitronic devices, similarly to single 22 the energy of IXs by U = −edFz. Therefore, although elec- electron transistors (SET) in semiconductor quantum dots. trically neutral, IXs couple to any gradient of Fz x;y paral- Clearly, for perspective devices, it is important to deter- lel to the planes.7,11 Such energy landscapes U x(;y )may be mine the behavior of IXs in different regimes, in order to generated by imperfections and defects, but they( can) also be optimize the performance. For example, large dipoles may engineered by properly designed split gates, as in traditional improve acceleration and operation rate, but ensuing smaller nano-electronics, to modulate the perpendicular field in the binding energies may cause dissociation and loss of the stored planes. Within their extended lifetime, IX can be moved over information. IXs are indeed polarizable quasi-particles, with both center-of-mass and internal dynamics. Since in typical systems, applied electrostatic fields in the CQW planes and a)Electronic mail: [email protected] b)Electronic mail: [email protected] the mutual Coulomb interaction of the pair are both in the few c)Electronic mail: [email protected] meV range, the quantum dynamics of IX under the influence 0021-9606/2015/142(3)/034701/12/$30.00 142, 034701-1 © 2015 AIP Publishing LLC This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 155.185.14.147 On: Thu, 15 Jan 2015 13:57:37 034701-2 Grasselli, Bertoni, and Goldoni J. Chem. Phys. 142, 034701 (2015) of accelerating potentials is a genuinely two-body process, and (Figure1). Accordingly, we consider the following electron- the internal quantum dynamics cannot be neglected in general. hole Hamiltonian: Although a fundamental problem, the quantum description of ˆ ˆ ˆ ˆ such two-body scattering process has been little discussed in H = T +UC +Uext; (1) 23 the literature. where In this paper we use a unitary propagation scheme to investigate theoretically the exact coherent dynamics of a single pˆ2 pˆ2 Tˆ = e + h ; (2) IX wavepacket describing scattering against different types of 2me 2mh electrostatic potential landscapes, namely, steps/downhills and 2 1 e = 4πϵ0 barriers/wells, using a minimal 1D model of an IX (Figure1). In UˆC = − ( ) ; (3) ϵ p 2 2 this sense, this is a two-particle generalization of the textbook r xˆe − xˆh + d ( ) description of one particle bouncing against potential steps and Uˆext = Ue xˆe; t +Uh xˆh; t : (4) barriers. However, since the electron and the hole reside in ( ) ( ) different layers, in typical devices, they are subject to different Here,p ˆe h ; xˆe h ; me h are the 1D linear momentum oper- ( ) ( ) ( ) potentials. Therefore, we have extensively investigated the ator, the coordinate operator, and the effective mass of the ˆ parameter space separately for the two particles, analyzing electron (hole), respectively. UC is the electron-hole Coulomb the time-dependent dynamics during scattering as well as the interaction in a medium of relative dielectric constant ϵ r. ˆ long-time behavior of the scattered wavepacket. In addition to Ue h xˆe h ;t represents the one-body electrostatic potential ( )( ( ) ) regions where IXs are reflected or transmitted almost as rigid which couples to the electron (hole) coordinatex ˆe h . Note ( ) objects, we identify experimentally relevant regimes where that in our model, the internal dynamics of the electron-hole scattering is a genuine quantum two-body process. In these pair is described in full, i.e., we do not assume any rigid 24,25 regimes, transmission or reflection of the wavepacket may be exciton approximation, nor the two particles are bound accompanied by excitation of the internal degrees of freedom, by construction: scattering against an external potentials may dissociation of the pair, or transmission in small periodic indeed result in excitation or dissociation of the pair. On the wavepackets due to dwelling of one particle in the barrier other hand, we neglect any lateral extension of the quantum region. states. Energy gaps induced by lateral confinement in a quan- In Sec.II, we set a 1D model of an IX and we briefly tum well or wire (tens of meV) far exceed, and therefore outline the numerical method used for propagation. In Sec. decouple from, the small binding energy of IXs (few meV).
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